Approximation of function using generalized Zygmund class
Abstract In this paper we review some of the previous work done by the earlier authors (Singh et al. in J. Inequal. Appl. 2017:101, 2017; Lal and Shireen in Bull. Math. Anal. Appl. 5(4):1–13, 2013), etc., on error approximation of a function g in the generalized Zygmund space and resolve the issue o...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-01-01
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| Series: | Advances in Difference Equations |
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| Online Access: | https://doi.org/10.1186/s13662-020-03197-5 |
| _version_ | 1818728426041769984 |
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| author | H. K. Nigam Mohammad Mursaleen Supriya Rani |
| author_facet | H. K. Nigam Mohammad Mursaleen Supriya Rani |
| author_sort | H. K. Nigam |
| collection | DOAJ |
| description | Abstract In this paper we review some of the previous work done by the earlier authors (Singh et al. in J. Inequal. Appl. 2017:101, 2017; Lal and Shireen in Bull. Math. Anal. Appl. 5(4):1–13, 2013), etc., on error approximation of a function g in the generalized Zygmund space and resolve the issue of these works. We also determine the best error approximation of the functions g and g ′ $g^{\prime }$ , where g ′ $g^{\prime }$ is a derived function of a 2π-periodic function g, in the generalized Zygmund class X z ( η ) $X_{z}^{(\eta )}$ , z ≥ 1 $z\geq 1$ , using matrix-Cesàro ( T C δ ) $(TC^{\delta })$ means of its Fourier series and its derived Fourier series, respectively. Theorem 2.1 of the present paper generalizes eight earlier results, which become its particular cases. Thus, the results of (Dhakal in Int. Math. Forum 5(35):1729–1735, 2010; Dhakal in Int. J. Eng. Technol. 2(3):1–15, 2013; Nigam in Surv. Math. Appl. 5:113–122, 2010; Nigam in Commun. Appl. Anal. 14(4):607–614, 2010; Nigam and Sharma in Kyungpook Math. J. 50:545–556, 2010; Nigam and Sharma in Int. J. Pure Appl. Math. 70(6):775–784, 2011; Kushwaha and Dhakal in Nepal J. Sci. Technol. 14(2):117–122, 2013; Shrivastava et al. in IOSR J. Math. 10(1 Ver. I):39–41, 2014) become particular cases of our Theorem 2.1. Several corollaries are also deduced from our Theorem 2.1. |
| first_indexed | 2024-12-17T22:29:48Z |
| format | Article |
| id | doaj.art-b5720bef28ee4c4dbf629df5d9ea43fb |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-17T22:29:48Z |
| publishDate | 2021-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-b5720bef28ee4c4dbf629df5d9ea43fb2022-12-21T21:30:13ZengSpringerOpenAdvances in Difference Equations1687-18472021-01-012021112210.1186/s13662-020-03197-5Approximation of function using generalized Zygmund classH. K. Nigam0Mohammad Mursaleen1Supriya Rani2Department of Mathematics, Central University of South BiharDepartment of Mathematics, Aligarh Muslim UniversityDepartment of Mathematics, Central University of South BiharAbstract In this paper we review some of the previous work done by the earlier authors (Singh et al. in J. Inequal. Appl. 2017:101, 2017; Lal and Shireen in Bull. Math. Anal. Appl. 5(4):1–13, 2013), etc., on error approximation of a function g in the generalized Zygmund space and resolve the issue of these works. We also determine the best error approximation of the functions g and g ′ $g^{\prime }$ , where g ′ $g^{\prime }$ is a derived function of a 2π-periodic function g, in the generalized Zygmund class X z ( η ) $X_{z}^{(\eta )}$ , z ≥ 1 $z\geq 1$ , using matrix-Cesàro ( T C δ ) $(TC^{\delta })$ means of its Fourier series and its derived Fourier series, respectively. Theorem 2.1 of the present paper generalizes eight earlier results, which become its particular cases. Thus, the results of (Dhakal in Int. Math. Forum 5(35):1729–1735, 2010; Dhakal in Int. J. Eng. Technol. 2(3):1–15, 2013; Nigam in Surv. Math. Appl. 5:113–122, 2010; Nigam in Commun. Appl. Anal. 14(4):607–614, 2010; Nigam and Sharma in Kyungpook Math. J. 50:545–556, 2010; Nigam and Sharma in Int. J. Pure Appl. Math. 70(6):775–784, 2011; Kushwaha and Dhakal in Nepal J. Sci. Technol. 14(2):117–122, 2013; Shrivastava et al. in IOSR J. Math. 10(1 Ver. I):39–41, 2014) become particular cases of our Theorem 2.1. Several corollaries are also deduced from our Theorem 2.1.https://doi.org/10.1186/s13662-020-03197-5Generalized Minkowski inequality (GMI)Best approximationGeneralized Zygmund classMatrix ( T ) $(T)$ meansC δ $C^{\delta }$ meansMatrix-Cesàro (δ order) ( T C δ ) $(TC^{\delta })$ |
| spellingShingle | H. K. Nigam Mohammad Mursaleen Supriya Rani Approximation of function using generalized Zygmund class Advances in Difference Equations Generalized Minkowski inequality (GMI) Best approximation Generalized Zygmund class Matrix ( T ) $(T)$ means C δ $C^{\delta }$ means Matrix-Cesàro (δ order) ( T C δ ) $(TC^{\delta })$ |
| title | Approximation of function using generalized Zygmund class |
| title_full | Approximation of function using generalized Zygmund class |
| title_fullStr | Approximation of function using generalized Zygmund class |
| title_full_unstemmed | Approximation of function using generalized Zygmund class |
| title_short | Approximation of function using generalized Zygmund class |
| title_sort | approximation of function using generalized zygmund class |
| topic | Generalized Minkowski inequality (GMI) Best approximation Generalized Zygmund class Matrix ( T ) $(T)$ means C δ $C^{\delta }$ means Matrix-Cesàro (δ order) ( T C δ ) $(TC^{\delta })$ |
| url | https://doi.org/10.1186/s13662-020-03197-5 |
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